Matrix representation of $\mathbb{C}$ 
Possible Duplicate:
Relation of this antisymmetric matrix $r = \begin{pmatrix} 0&amp;1\\ -1&amp;0 \end{pmatrix}$ to $i$ 

Let $H$ be the subset of $M_2(\mathbb R)$ consisting of all matrices of the form  $\begin{pmatrix}a & -b \\ b & a\end{pmatrix}$ for $a, b \in \mathbb R$. 


*

*Show that $(\mathbb C,+)$ is isomorphic  to $(H,+)$.

*Show that $(\mathbb C, \times)$ is isomorphic to $(H, \times)$.


$H$ is said to be a matrix representation of the complex numbers.
I beg some help please. I fail even to define one to one functions mapping $\mathbb C$ onto $H$. All the best.
 A: Our matrices are of the form $$\left(\begin{smallmatrix} a & -b \\ b & a\end{smallmatrix}\right)$$ While our complex numbers are of the form $a + bi$ 
Both of these depend on $a, b$ 
Can you see a way to map from $\mathbb{C} \rightarrow H$ ? 
Hint: For the harder one, multiplication $((a+bi)(c+di)) = (ac +adi + cbi -bd)$ so to have an isomorphism we will want $f(ac - bd + (ad + cb)i) = f((a+bi))f((c+di))$ 
we'll try the only choice that really makes sense $a + bi \mapsto \left(\begin{smallmatrix} a& -b \\ b & a\end{smallmatrix}\right)$
Then $$f(ac - bd + (ad + cb)i) = \left(\begin{smallmatrix} (ac - bd) & (-ad -cb) \\ (ad + cb) & (ac - bd)\end{smallmatrix}\right)= \left(\begin{smallmatrix} a & - b \\ b & a\end{smallmatrix}\right)\left(\begin{smallmatrix} c & -d \\ d & c\end{smallmatrix}\right) = f(a + bi)f(c + di)$$ 
Now of course, you have to show that this is one-to-one and onto (although that shouldn't be that hard) and I believe the addition should be similar. 
A: Here's a hint : If you consider a complex number $z$, it can be written as $a+bi$ with $a,b \in \mathbb R$. (I haven't chosen the letters $a$ and $b$ for no reason.) 
If you want a proof I don't mind showing. Just ask.
Hope that helps,
